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Purely Extending Modules and Their Generalizations

  • Shiv Kumar (Department of Mathematical Sciences, IIT(B. H. U.) ) ;
  • Ashok Ji Gupta (Department of Mathematical Sciences, IIT(B. H. U.) )
  • Received : 2021.06.15
  • Accepted : 2022.12.06
  • Published : 2023.03.31

Abstract

A purely extending module is a generalization of an extending module. In this paper, we study several properties of purely extending modules and introduce the notion of purely essentially Baer modules. A module M is said to be a purely essentially Baer if the right annihilator in M of any left ideal of the endomorphism ring of M is essential in a pure submodule of M. We study some properties of purely essentially Baer modules and characterize von Neumann regular rings in terms of purely essentially Baer modules.

Keywords

Acknowledgement

We are very thankful to the referee for a thorough report and for many helpful suggestions. The first author also gratefully acknowledges financial support from UGC, INDIA to carry out this research work.

References

  1. S. E. Atani, M. Khoramdel and S. D. Pishhesari, Purely Baer Modules and Purely Rickart Modules, Miskolc Math. Notes, 19(1)(2018), 63-76.  https://doi.org/10.18514/MMN.2018.1484
  2. G. F. Birkenmeier, J. K. Park and S. T. Rizvi , Extensions of Rings and Modules, Research Monograph, Birkhauser/Springer, (2013). 
  3. A. Chatters and C. R. Hajarnavis, Rings in which every complement right ideal is a direct summand, Quart. J. Math. Oxford Ser., 28(1)(1977), 61-80.  https://doi.org/10.1093/qmath/28.1.61
  4. A. W. Chatters and S. M. Khuri, Endomorphism Rings of Modules over Nonsingular CS Rings, J. London Math. Soc., 2(3)(1980), 434-444.  https://doi.org/10.1112/jlms/s2-21.3.434
  5. J. Clark, On Purely Extending Modules, Proceedings of the International Conference in Dublin, August 10-14, (1998), (Basel Birkhauser: Trends in Mathematics), 353-358. 
  6. P. M. Cohn, On Free Product of Associative Rings, Math. Z., 71(1)(1959), 380-398.  https://doi.org/10.1007/BF01181410
  7. D. J. Fieldhouse, Pure Theories, Math. Ann., 184(1970), 1-18.  https://doi.org/10.1007/BF01350610
  8. K. R. Goodearl, Von Neumann Regular Rings, Monographs and Studies in Maths, 4, Pitman London(1979). 
  9. T. Y. Lam , Lectures on Modules and Rings, Graduate Texts in Mathematics, 189, Springer, New York(1999). 
  10. G. Lee, S. T. Rizvi and C. S. Roman, Modules whose Endomorphism Rings are Von Neumann Regular, Comm. Algebra, 41(2013), 4066-4088.  https://doi.org/10.1080/00927872.2012.700979
  11. S. H. Mohamed and B. J. Muller, Continuous and Discrete Modules, London Math. Soc. Lecture Notes, 147, Cambridge Univ. Press (1990). 
  12. T. H. N. Nhan, Essentially Baer Modules, Chebyshevskii Sb., 16(3)(2015), 355-375. 
  13. S. T. Rizvi and C. S. Roman, Baer and quasi-Baer Modules, Comm. Algebra, 32(1)(2004), 103-123.  https://doi.org/10.1081/AGB-120027854
  14. A. Tercan and C. C. Yucel, Module theory, Extending modules and generalizations, Birkhauser Basel, Springer, Switzerland(2016). 
  15. A. K. Tiwari and S. A. Paramhans, On Closures of Submodules, Indian J. Pure Appl. Math., 8(1977), 1415-1419. 
  16. B. Ungor and S. Helicioglu, Strongly Extending Modules, Hacet. J. Math. Stat., 42(5)(2013), 465-478. 
  17. R. Wisbaur, Foundations of Module and Ring theory, A handbook for study and research, Philadelphia(1991).